Simultaneous inversion of nmr multiple echo trains and conventional logs

ABSTRACT

A method for estimating one or more properties as a function of depth of an earth formation penetrated by a borehole includes: receiving nuclear magnetic resonance (NMR) logging data having NMR echo trains as a function of depth in the borehole; receiving non-NMR logging data having non-NMR measurement values for one or more types of non-NMR measurements as a function of depth in the borehole; generating an evolution matrix (E) representing a mathematical relationship between the one or more properties in property matrix (P) to be estimated and the NMR logging data and non-NMR logging data matrix (M); generating a matrix equation of the form M=E·P; and inverting the matrix equation to estimate the one or more properties as a function of depth.

BACKGROUND

Earth formations may be used for various purposes such as hydrocarbonproduction, geothermal production and carbon dioxide sequestration. Inorder to efficiently use the formations, measurements are typicallyperformed on the formations using sensors or tools disposed in boreholespenetrating the formations.

Measurement data from a sensor may be inverted using an inversionalgorithm in order to estimate parameters of a mathematical model of theearth formation from which the measurement data was obtained. That is,the mathematical model with accurate parameters will provide the same ornear measurement data when measurements on the mathematical model of theearth formation using the same type of sensor are simulated. Hence, itwould be well received in the drilling industry if inversion algorithmsand methods were developed to invert sensor data in order to moreaccurately estimate parameters of a mathematical model that representsan earth formation of interest.

BRIEF SUMMARY

Disclosed is a method for estimating one or more properties as afunction of depth of an earth formation penetrated by a borehole. Themethod includes: receiving with a processor nuclear magnetic resonance(NMR) logging data comprising NMR echo trains as a function of depth inthe borehole; receiving with the processor non-NMR logging datacomprising non-NMR measurement values for one or more types of non-NMRmeasurements as a function of depth in the borehole; generating with theprocessor an evolution matrix (E) representing a mathematicalrelationship between the one or more properties in property matrix (P)to be estimated and the NMR logging data and non-NMR logging data matrix(M); generating with the processor a matrix equation of the form M=E·P;and inverting with the processor the matrix equation to estimate the oneor more properties as a function of depth.

Also disclosed is a system for estimating one or more properties as afunction of depth of an earth formation penetrated by a borehole. Thesystem includes: a carrier configured to be conveyed through theborehole; a nuclear magnetic resonance (NMR) logging tool disposed onthe carrier and configured to provide NMR logging data having NMR echotrains as a function of depth in the borehole; a non-NMR tool disposedon the carrier and configured to provide non-NMR logging data havingnon-NMR measurement values for one or more types of non-NMR measurementsas a function of depth in the borehole; and a processor. The processoris configured to: receive the NMR logging data; receive the non-NMRlogging data; generate an evolution matrix (E) representing amathematical relationship between the one or more properties (P) to beestimated and the NMR logging data and non-NMR logging data (M);generate a matrix equation of the form M=E·P; and invert the matrixequation to estimate the one or more properties as a function of depth.

BRIEF DESCRIPTION OF THE DRAWINGS

The following descriptions should not be considered limiting in any way.With reference to the accompanying drawings, like elements are numberedalike:

FIG. 1 is a cross-sectional view of an embodiment of a bottomholeassembly (BHA) disposed in a borehole penetrating the earth;

FIG. 2 is a cross-sectional view of hydrocarbon production apparatusconfigured to perform one or more physical actions related to theproduction of hydrocarbons using the one or more formation properties asestimated by the methods disclosed herein;

FIG. 3 presents is a flow chart depicting aspects of using known fluidproperties, NMR measurements of multiple echo-trains and non-NMRmeasurements for simultaneous one-dimensional inversion to estimate oneor more properties of an earth formation upon which the NMR and non-NMRmeasurements were performed;

FIG. 4 presents an overview of an NMR inversion process;

FIG. 5 depicts aspects of mathematical representations of transversalmagnetization (M) evolution for single and multiple experiments andsingle and multiple fluids;

FIG. 6 depicts aspects of NMR forward modeling responses due to NMRacquisition sequences;

FIG. 7 depicts aspects of NMR T2 intrinsic spectra for each of oil, gasand water obtained from inversion;

FIG. 8 depicts aspects of accounting for non-NMR measurements in theinversion;

FIG. 9 depicts aspects of twenty four NMR experiments with differentacquisition parameters performed by an NMR tool; and

FIG. 10 is a flow chart for a method for estimating one or moreproperties as a function of depth of an earth formation penetrated by aborehole.

DETAILED DESCRIPTION

A detailed description of one or more embodiments of the disclosedapparatus and method presented herein by way of exemplification and notlimitation with reference to the figures.

Generally speaking, a sought-after petrophysical parameter obtained fromNMR (nuclear magnetic resonance) logs is the lithology independentporosity. NMR porosity is not related to lithology but it is related tothe fluids Hydrogen Index. By employing simultaneous inversion it ispossible to get the porosity corrected by hydrogen index.

All of the inversion processes are very ill conditioned least squareproblems. It is generally always necessary to use regularization to getstable solutions and the amount of necessary regularization is noisedependent. Normally results are good only in large porosities.

In low porosities as in tight sand and shale, the porosity fromconventional logs is not well defined because it is always obtained as adifference between the measured parameter and a matrix parameter. Whenthe porosity approaches to zero, the difference approaches to zero too,the errors in the porosity trend to infinite. The error is not onlyrelated to measure errors, it is also related to the error in theknowledge of the matrix parameter. NMR porosity errors can be decreasedby reducing the acquisition noise and they are not related to the matrixparameters. NMR is the only technology able to get reliable porositywhen the porosity is low or when matrix parameters are not well known.

Disclosed are methods, which may be implemented as algorithms by aprocessor, for inversion of measurement data obtained from downholetools. The methods call for simultaneous inversion of measurement dataobtained from a downhole nuclear magnetic resonance (NMR) tool and fromother types of downhole tools (i.e., non-NMR tools) known in the art.The simultaneous inversion of these types of measurement data providesmore accurate estimates of one or more properties of an earth formationfrom which the measurement data was obtained than conventional inversionmethods. Each of the properties of interest has a defined mathematicalrelationship with an NMR measurement value and one or more types ofnon-NMR measurement values. The multiple mathematical relationships foreach property in the simultaneous inversion provides for the increase inaccuracy of the estimates of the one or more properties of interest bysimultaneously satisfying multiple independent constraints. The use ofconventional logs in penalty equations in place of or in addition toregularization is disclosed to improve quantification of fluids inone-dimensional (1D) inversion. Accordingly, the quantification offluids is improved and a more accurate porosity value can be computed.If the fluids in the reservoir are known, then one-dimensional inversionis normally used to quantify the fluid volumes in the NMR measured zone.If the fluids are unknown, then two-dimensional and/or three-dimensionalinversion are used to typify and quantify fluids. Parameters related tofluids of interest include T2 (normally a spectrum of values), diffusionconstant D and T1/T2 ratio, R. If R and D are known, T2 spectrum is justthe unknown. This inversion is called 1D inversion. If R or D is known,the unknowns will be T2-D or T2-R. This inversion is called 2D inversionand the output is a spectrum in two dimensions, which may be presentedas a map. If T2, R and D are unknowns, then the inversion is called 3Dinversion and the output is a three dimensional spectrum.

FIG. 1 is a cross-sectional view of an embodiment of an NMR tool 10 anda non-NMR tool 15 disposed in a borehole 2 penetrating the earth 3,which includes an earth formation 4. The downhole tools 10 and 15 areconveyed through the borehole 2 by a carrier 5, which can be an armoredwireline 6. Besides supporting and conveying the downhole tools 10 and15 in the borehole 2, the armored wireline 6 can include electricalconductors for conveying electrical signals between the downhole tools10 and/or 15 and a surface receiver such a computer processing system 12in real time. A rig 8 is configured to conduct borehole-relatedoperations such as conveying the carrier 5 through the borehole 2 as anon-limiting example.

The NMR tool 10 provides an NMR log of data as a function of depth. Thenon-NMR tool 15 represents one or more downhole non-NMR sensing toolsthat are configured to provide other non-NMR types of sensing log dataas a function of depth. Downhole electronics 11 are configured tooperate the NMR tool 10, operate the one or more non-NMR downholesensing tools 15, process measurement data obtained downhole from any ofthe downhole tools, and/or act as an interface with telemetry tocommunicate data or commands between downhole components and thecomputer processing system 12 disposed at the surface of the earth 3.System operation and data processing operations may be performed by thedownhole electronics 11, the computer processing system 12, or acombination thereof. In the embodiment of FIG. 1, the downhole tools 10and 15 are side tools having components (not shown) configured to urgethe tools 10 and 15 to a side of the borehole 2 for performingmeasurements. In an alternative embodiment, the downhole tools 10 and 15are centralized tools having components such as a centralizer (notshown) configured to urge the tools 10 and 15 to the center of theborehole 2 for performing measurements.

The NMR tool 10 is configured to perform NMR measurements on theformation 4. NMR measurements are performed in a volume of interest 9.This volume may be torus-shaped, surrounding the NMR tool 10, or, whenusing a side-looking NMR tool, may be on one side only. The NMRmeasurements may yield a longitudinal relaxation time constant T₁ and atransverse relaxation time constant T₂ (or distributions thereof, seebelow). T₁ relates to a time that is characteristic of the amount oftime required for magnetic polarization of the hydrogen atoms in thevolume of interest. In general, longer wait times (TW) provide moremagnetic polarization than shorter wait times. T₂ relates to anexponential decay time constant that corresponds to a characteristic orproperty of the formation 4 material. Transverse relaxation relates tothe irreversible loss of phase coherence of individual hydrogen nuclei(=protons) in the formation 4 material while precessing about a staticmagnetic field during an NMR measurement. There is not one single valueof T₂ for formation rock but a wide distribution of values lyinganywhere between fractions of a millisecond (ms) and several seconds forexample. The distributions of T₁ and T₂ values are principal outputs ofthe NMR tool 10 and together may be referred to as an NMR log.Components in the NMR tool 10 include a static magnetic field source 13that magnetizes formation materials and an antenna 14, which mayrepresent one or more antennas, which transmit precisely timed bursts ofradio-frequency energy (e.g., a CPMG sequence) that provides anoscillating magnetic field. In a time period between these pulses, theantenna receives a decaying echo signal from those protons that are inresonance with the static magnetic field produced by the static magneticfield source. Because a linear relationship exists between the protonresonance frequency and the strength of the static magnetic field, thefrequency of transmitted radio-frequency energy can be tuned toinvestigate volumes of interest having different diameters around ordistances from the NMR tool 10. It can be appreciated that the NMR tool10 may include a variety of components and configurations as known inthe art of NMR. It can be appreciated that the NMR tool 10 may becalibrated to a known micro-porosity and/or other known properties of asubsurface material by analysis or by testing in field or laboratoryconditions using subsurface materials having a known micro-porosityand/or other known properties. In that NMR tools are known in the art,specific details of components and configurations of these tools are notdiscussed in further detail.

The one or more non-NMR downhole sensing tools 15 are configured tosense properties of the formation 4 using principles that are differentfrom NMR. Non-limiting embodiments of the one or more non-NMR downholesensing tools 15 include radiation bombardment tools that bombard theformation with neutrons and/or gamma rays and detect the resultingradiation in order to estimate density or porosity, natural gamma-raytools that detect the natural gamma-rays emitted by a formation,acoustic tools that measure the acoustic impedance of a formation, andresistivity tools that measure the resistivity or conductivity of aformation. Gamma-ray logging tools bombard the formation with from achemical source of gamma-rays. Radiation is scattered back to thelogging tool with an intensity dependent on the electron density of theformation material. The density of the formation material can then beextracted from the amplitude of the back-scattered radiation (e.g.,gamma-rays). It can be appreciated that the one or more downhole sensingtools 15 may be calibrated to a known property and/or other knownproperties of a subsurface material by analysis or by testing in fieldor laboratory conditions using subsurface materials having a knowncorresponding property. In that these types of downhole sensing toolsare known in the art, specific details of components and configurationsof these tools are not discussed in further detail.

When data from the one or more non-NMR downhole tools 15 are used incombination with the data from the NMR tool 10 for inversion, thecombined data inversion provides a more complete and accuratecharacterization of the formation 4 than would be possible with any onetype of data alone.

FIG. 2 is a cross-sectional view of hydrocarbon production apparatus 20configured to perform one or more physical actions related to theproduction of hydrocarbons using the one or more formation properties asestimated by the methods disclosed herein. The hydrocarbon productionapparatus 20 may include a downhole production tool 21 that isconfigured to be conveyed through the borehole 2 by a carrier, such asan armored wireline, to perform a downhole physical action related tothe production of hydrocarbons. In one or more embodiments, the downholeproduction tool 21 is a perforation tool 22 that is configured toperforate a casing 23 lining the borehole 2. The casing 23 is perforatedat a depth or range of depths as determined by the one or more estimatedformation properties for the efficient production of hydrocarbons. Thatis, the depth or range of depths is selected to produce hydrocarbonswhile excluding the production of water. The hydrocarbon productionapparatus 20 may include a hydraulic fracturing system 24. The hydraulicfracturing system 24 is configured to hydraulically fracture theformation 4 at a depth or range of depths as determined by the one ormore estimated formation properties. The hydrocarbon productionapparatus 20 may include other downhole production tools and/orproduction systems not shown. The hydrocarbon production apparatus 20may include a controller 25 configured to accept inputs derived from theone or more estimated formation properties and output a control signalto tools and/or systems for producing hydrocarbons in order to controlthe tools and/or systems in accordance with one or more estimatedformation properties.

FIG. 3 presents is a flow chart depicting aspects of using known fluidproperties, NMR measurements of multiple echo-trains and non-NMRmeasurements for simultaneous 1D inversion to estimate one or moreproperties of an earth formation upon which the NMR and non-NMRmeasurements were performed. Non-limiting embodiments of the one or moreproperties include porosity, water volume, oil volume, gas volume,permeability, fluid viscosity, grain size and capillary pressure. Inthat the mathematical relationships between each of the one or moreproperties to be estimated and the known fluid parameters, NMRmeasurements and/or non-NMR measurements are known in the art, they arenot discussed in further detail. In FIG. 3, Rxo refers to shallowresistivity measurements on the order of less than two inches. Rt refersto deeper resistivity measurements on the order of several feet. Fromthe petrophysical analysis, an amount of hydrocarbons in place andproductivity of the formation can be estimated. With this information,well completion tasks can be performed such as determining where toperforate a casing lining the well in order to produce hydrocarbonswhile excluding the production of water.

FIG. 4 presents an overview of an NMR inversion process. The NMR-CPMGpulse sequence depicted in the upper-left corner results in the measuredecho train (M(t)) depicted in the upper-right corner. When the echotrain (M(t)) is inverted in accordance with the equation depicted in thelower-right corner where ε(t) is a noise term (fundamentally thermalnoise in a receiver antenna, the NMR T2 spectrum (F(T2)) depicted in thelower-left corner is obtained.

FIG. 5 depicts aspects of mathematical representations of transversalmagnetization (M) evolution for single and multiple experiments andsingle and multiple fluids.

FIG. 6 depicts aspects of NMR forward modeling responses in the lowerpart of the figure for the NMR acquisition sequences depicted in theupper part of the figure. The experiment parameters for each of thesequences are presented in Table 2 below.

FIG. 7 depicts aspects of NMR T2 intrinsic spectra (i.e., T2 spectra foreach of oil, gas and water) obtained from inversion. The equation to beinverted is of the form M=E·P+ε where each of the terms is representedby a matrix. Also depicted are embodiments of a least squares solutionand a weighted and regularized least squares solution. The forward modelis mathematically represented by a forward matrix. When the model istime related, the matrix may be referred to as an evolution matrix.

FIG. 8 depicts aspects of accounting for non-NMR measurements in theevolution matrix E. In this case, the magnetization matrix M alsoincludes the non-NMR measurements.

Methods for simultaneous inversion of NMR multiple echo trains andconventional logs are discussed in more detail next. Currently the NMRinversion is typically done using only NMR data and petrophysicalanalysis takes the inversion output as raw data. Most of the time thevertical resolution does not match or some NMR results as CBW, BVI orPHE are taken as true when in fact they may be different. The NMR depthof investigation is similar to the porosity conventional tools but it isnot similar to resistivity tools. Parameters of fluids will be necessaryfor 1D inversion and matrix parameters will be necessary for penaltyequations.

Vertical resolution is discussed next. The NMR vertical resolution mustbe matched to the conventional (i.e., non-NMR) logs. In the currentinversion process the echo trains are stacked until the “CHI” is below 2pu. That is a good procedure considering only NMR data. The same CLS(constrained least squares) eleven points filter used in conventionallogs will be applied to the echoes in place of stacking. The desired CHImust be achieved selecting the proper logging speed. The normal weightsof the CLS eleven points filter are in Table 1. A stacking of twosamples will be necessary for echoes with alternating phase.

TABLE 1 −5 −4 −3 −2 −1 0 1 2 3 4 5 0.0059 0.0298 0.0735 0.1266 0.17040.1875 0.1704 0.1266 0.0735 0.0298 0.0059

NMR Tool response and inversion of a single echo train are discussednext. NMR T2 spectrum, from now on F(T2), is a rock property dependentof the experiment wherein different experiment parameters producedifferent spectra. Standard spectra are represented in a semi log graphusing log(T2) as the x-axis. From now on all integral equations along T2are in fact along log(T2).

For each differential of T2 (see FIG. 3), the NMR tool responds with atime exponential function de(t). Equation 1 represents the wholespectrum of the tool response. The tool makes a transform from the T2domain to time domain through a pulse sequence as illustrated in FIG. 4.

$\begin{matrix}{{{e(t)} = {{\int_{0}^{\infty}{{{F\left( {T\; 2} \right)} \cdot {\exp \left( {- \frac{t}{T\; 2}} \right)} \cdot {dT}}\; 2}} + {\varepsilon (t)}}}{{e(t)} = {\mathcal{L}\left\lbrack {F\left( {T\; 2} \right)} \right\rbrack}}} & (1)\end{matrix}$

The term ε(t) is the measure noise, fundamentally thermal noise in theantenna. It can be seen that L[F(T2)] is a linear transform shownEquation 2.

[α·

(

2)+

·H(

2)]=α·

[

(

2)]+

·

[H(

2)]  (2)

Using a sum of nB weighted exponential functions exp(−t/T_(2i)) withfixed time constant T_(2i) to approximate the echo train, the weightsfound B_(i) with i∈(1, . . . , nB) can be used as weights of nBfunctions of T2 to get the spectrum from the echo train.

If exponential functions are used in time domain, delta Dirac functionsmust be used in T2 domain because L[δ(T2−T_(2i))]=exp(−t/T_(2i)).

Solving the inversion of a single echo train with nE echoes consist into solve nE linear equations (see Equations 3).

                   (3) $\left\{ \begin{matrix}{B_{1}{\exp \left( {{{- t_{1}}/T}\; 2_{1}} \right)}} & + & \ldots & + & {B_{n}{\exp \left( {{{- t_{1}}/T}\; 2_{n}} \right)}} & + & \ldots & + & {B_{nB}{\exp \left( {{{- t_{1}}/T}\; 2_{nB}} \right)}} & = & E_{1} \\\ldots & \; & \; & \; & \ldots & \; & \; & \; & \ldots & = & \cdot \\\ldots & \; & \; & + & {B_{n}{\exp \left( {{{- t_{i}}/T}\; 2_{n}} \right)}} & + & \; & \; & \ldots & = & E_{i} \\\ldots & \; & \; & \; & \ldots & \; & \; & \; & \ldots & = & \cdot \\{B_{1}{\exp \left( {{{- t_{nE}}/T}\; 2_{1}} \right)}} & + & \ldots & + & {B_{n}{\exp \left( {{{- t_{nE}}/T}\; 2_{n}} \right)}} & + & \ldots & + & {B_{nB}{\exp \left( {{{- t_{nE}}/T}\; 2_{nB}} \right)}} & = & E_{nE}\end{matrix} \right.$

This is a very ill conditioned system. As an example, for nE=500; nB=28and TE=0.6 the condition number of the matrix A in Equation 4 is1.2592e+016.

$\begin{matrix}{{{A = \begin{bmatrix}{\exp \left( {{{- t_{1}}/T}\; 2_{1}} \right)} & \ldots & {\exp \left( {{{- t_{1}}/T}\; 2_{n}} \right)} & \ldots & {\exp \left( {{{- t_{1}}/T}\; 2_{nB}} \right)} \\\ldots & \; & \ldots & \; & \ldots \\\ldots & \; & {\exp \left( {{{- t_{i}}/T}\; 2_{n}} \right)} & \; & \ldots \\\ldots & \; & \ldots & \; & \ldots \\{\exp \left( {{{- t_{nE}}/T}\; 2_{1}} \right)} & \ldots & {\exp \left( {{{- t_{nE}}/T}\; 2_{n}} \right)} & \ldots & {\exp \left( {{{- t_{nE}}/T}\; 2_{nB}} \right)}\end{bmatrix}};}\mspace{79mu} {{B = \begin{bmatrix}B_{1} \\\vdots \\B_{n} \\\vdots \\B_{nB}\end{bmatrix}};{E = \begin{bmatrix}E_{1} \\\vdots \\E_{i} \\\vdots \\E_{nE}\end{bmatrix}}}} & (4)\end{matrix}$

Solving the system require regularization to get stable solutions. Usingminimum norm regularization with a factor β and least squares, thesolution is in Equation 5.

B=(A ^(T) ·A+β ² ·I)⁻¹ ·A ^(T) ·E  (5)

The echo train approximation is given by Equation 6 and an example is inFIG. 5.

FIT=A·B  (6)

Inversion of multiple echo trains is discussed next. Current NMR dataacquisitions may have several echo trains with different acquisitionparameters. In one or more embodiments, they may have one echo trainwith the shorter available TE (inter-echo time) and short TW (wait time)for CBW and other echo train with long TW to get the whole spectrum. Forexample, an NMR tool with a gas acquisition mode may perform twenty fourexperiments with different acquisition parameters as illustrated in FIG.9 with respect to the acquisition parameters listed in Table 2. NE isthe number of echoes acquired. TE is the time between echoes. TW is thetime between the last echo of one experiment to the first one of thenext experiment. Frequency is in MHz.

TABLE 2 FREQ TE [μSec] TW [mSec] nE 2A 961.77 600 8302 690 2C 961.77 40020 25 2B 961.77 600 983 690 2C 961.77 400 20 25 2D 961.77 400 50 25 2E961.77 400 100 25 4A 858.52 600 8302 690 4C 858.52 400 20 25 4B 858.52600 983 690 4C 858.52 400 20 25 4D 858.52 400 50 25 4E 858.52 400 100 251A 767.77 600 9222 690 1C 767.77 400 20 25 1B 767.77 600 983 690 1C767.77 400 20 25 3A 688.38 600 9222 690 3C 688.38 400 20 25 3B 688.38600 983 690 3C 688.38 400 20 25 5A 613.72 600 11097 690 5C 613.72 400 2025 6A 555.27 600 11097 25 6C 555.27 400 20 25

The main objectives of the multiple echo train inversion are typify andquantify fluids in the measured zone and to use the volume of fluids tocorrect the porosity by hydrogen index. The challenge is to get stableand reliable solutions. As disclosed herein, stable and reliablesolutions are obtained by inverting the echo train twice. The first timeto get the apparent T2 spectrum and the second time to get one intrinsicspectrum for each fluid, using the apparent spectrum obtained in thefirst run as penalty equations, in place of normal regularization. Toinclude conventional logs in the inversion, it is necessary to add thematrix volume as an unknown and an equation where weight·Σunknowns=weight with a large weight.

Computing apparent T2 spectrum is discussed next. In a multiple echotrain acquisition, there are experiments with different G·TE anddifferent TW. To invert echo trains with different G·TE together, onlyone fluid will be considered present with a diffusion constant near tothe diffusion constant of water.

To invert echo trains with different TWs, joint inversion will be used.Unknowns corresponding to a T2<TW/3 for each echo train will be sharedby all the echo trains. The space of unknowns will be extended untilunknowns corresponding to T2=3·TW. Unknowns corresponding toTW/3<T2<3·TW will be spaced twice than ones corresponding to T2<TW/3 andbeing auxiliary solutions.

In the example of FIG. 8, there are experiments with TW=20, TW=50,TW=100 and TW=983. Choosing 0.5/√2 as first bin and 28 bins total, theexponential base functions have time constants as given in Table 3, thebracketed T2 are in the range TW/3<T2<3·TW. The signals for this T2range are not totally polarized and the corresponding functions beingbase of auxiliary unknowns.

To resolve all experiments together, a data vector is createdconcatenating all echo trains, Equation 7. Assuming k experiments, eachof them has an echo train Ej, j∈(1, . . . , k) with nEj echoes. The datavector thus created has Σ_(j=1) ^(k)nE_(j) elements.

$\begin{matrix}{E = \begin{bmatrix}E_{1} \\\vdots \\E_{j} \\\vdots \\E_{k}\end{bmatrix}} & (7)\end{matrix}$

The unknowns will be divided in main and auxiliary. The main unknownswill be share by all experiments in the range T2<TW/3. The auxiliaryones will be share by experiments with same TW in the rangeTW/3<T2<3·TW.

The main unknowns are defined by the first bin, the increment and numberof bins. In the example of FIG. 8 with respect to Table 3, the mainunknowns in the first column have a first bin at T2=0.35, an incrementof √2 and 28 bins. The first bin and the number of bins may be selectedto cover all possible T2, Equation 8.

$\begin{matrix}{B = \begin{bmatrix}B_{1} \\\vdots \\B_{nB}\end{bmatrix}} & (8)\end{matrix}$

It will require many sets of auxiliary unknowns as there different shortTWs. Reducing the number of unknowns in a twofold increment is electedfor them, Equation 9.

$\begin{matrix}{{B_{m} = \begin{bmatrix}B_{m_{1}} \\\vdots \\B_{m_{nBm}}\end{bmatrix}}{m \in \left( {1,\ldots \mspace{11mu},{nTWS}} \right)}{{nTWS} = {\# \mspace{14mu} {short}\mspace{14mu} {TWs}}}} & (9)\end{matrix}$

A vector of extended unknowns is created concatenating the main unknownsand all set of auxiliary ones, Equation 10.

$\begin{matrix}{B^{*} = \begin{bmatrix}B \\B_{1} \\\vdots \\B_{nTWS}\end{bmatrix}} & (10)\end{matrix}$

Posing a least squares problem, Equation 11, the forward matrix resultsas Equation 12, all 0 are zero matrices with the necessary dimension tofill A.

The A_(l) sub matrices are created as Equation 13. The term nBm is thenumber of bins of the partial polarized experiment m such that thecorresponding T2 are less than TW/3.

$\begin{matrix}{{A_{i} = \begin{bmatrix}{{\exp \left( {{{- t_{1}^{l}}/T}\; 2_{1}} \right)} \cdot {D\left( t_{1}^{l} \right)}} & \cdots & {{\exp \left( {{{- t_{1}^{l}}/T}\; 2_{n}} \right)} \cdot {D\left( t_{1}^{l} \right)}} & \ldots & {{\exp \left( {{{- t_{1}^{l}}/T}\; 2_{{nB}_{m}}} \right)} \cdot {D\left( t_{1}^{l} \right)}} \\\ldots & \; & \ldots & \; & \ldots \\\cdots & \; & {{\exp \left( {{{- t_{i}^{l}}/T}\; 2_{n}} \right)} \cdot {D\left( t_{i}^{l} \right)}} & \; & \ldots \\\cdots & \; & \ldots & \; & \cdots \\{{\exp \left( {{{- t_{nE}^{l}}/T}\; 2_{1}} \right)} \cdot {D\left( t_{nE}^{l} \right)}} & \cdots & {{\exp \left( {{{- t_{nE}^{l}}/T}\; 2_{n}} \right)} \cdot {D\left( t_{nE}^{l} \right)}} & \ldots & {{\exp \left( {{{- t_{nE}^{l}}/T}\; 2_{{nB}_{m}}} \right)} \cdot {D\left( t_{nE}^{l} \right)}}\end{bmatrix}}\mspace{20mu} {{D\left( t_{i}^{l} \right)} = {\exp \left( {\frac{{- \gamma^{2}} \cdot D_{w}^{2} \cdot \left( {{G_{f}^{2} \cdot {TE}_{f}^{2}} - {G_{l}^{2} \cdot {TE}_{l}^{2}}} \right)}{12} \cdot t_{i}^{l}} \right)}}} & (13)\end{matrix}$

The D_(l) sub matrices have the same structure than Equation 13 but T2are restricted to TW/3<T2<3·TW.

Weighting is discussed next. To take account the difference of noisebetween experiments, a weighting matrix W is applied to Equation 11. Wis a diagonal matrix with the elements equal to the inverse of power twoof the standard deviation of each experiment, Equation 14.

$\begin{matrix}{W = \begin{bmatrix}{\sigma_{1}^{- 2} \cdot 1_{1}} & 0 & 0 & 0 & 0 \\0 & \cdot & 0 & 0 & 0 \\0 & 0 & {\sigma_{j}^{- 2} \cdot 1_{j}} & 0 & 0 \\0 & 0 & 0 & \cdot & 0 \\0 & 0 & 0 & 0 & {\sigma_{k}^{- 2} \cdot 1_{k}}\end{bmatrix}} & (14)\end{matrix}$

1_(j) is the identity matrix with size (1_(j))=nE_(j) and σ_(j) is thestandard deviation of the channel Y after phase rotation.

Regularization and final result are discussed next. Different testsshowed that the most stable results are obtained regularizing only themain unknowns using minimum norm regularization. Changing the identitymatrix leaving ones only in places corresponding to the main unknownsand calling 1* to the modified identity matrix, the solution of theweighted and regularized least squares problem is Equation 15.

B*=(A ^(T) ·W·A+β ²·1*)⁻¹ ·A ^(T) ·W·E; B* _(j)≧0, ∀j  (15)

Assuming k experiments and calling STi to the first element of theexperiment i, the fit of all experiments is Equation 16. The fit of eachecho train are sub arrays of FIT, the fit of the echo train i isFIT_(i)=FIT(ST_(i):ST_(i)+nE_(i)−1).

FIT=A·B*  (16)

For each i experiment, i∈(1, . . . , k), a CHI_(i) can be computed asequation 17.

$\begin{matrix}{{CHI}_{i} = \frac{\sum\limits_{j = {ST}_{i}}^{{ST}_{i} + {nE}_{i} - 1}\left( {E_{j} - {FIT}_{j}} \right)^{2}}{{nE}_{i}}} & (17)\end{matrix}$

The first nB elements of B* are the T2 apparent spectrum PP, Equation18.

PP=B*(1:nB)  (18)

Computing intrinsic and hydrogen index (HI) corrected T2 spectrumassuming known fluids is discussed next. Knowing the type of fluids, theparameters of them can be estimated in function of the fluids property,pressure and temperature. These fluid parameters are used to construct aforward matrix that is able to resolve the T2 spectrum in three separatespectra, one for water, one for oil and other for gas. The sum of threeis the intrinsic and HI corrected T2 because D_(w,o,g), HI_(w,o,g) andR_(w,o,g) are known and included in the forward matrix.

The model includes all parameters is in equation 19. T2_(j) is the T2corresponding to the unknown j, time=n·TE, k number of experiments,TE_(i) inter echo time of the experiment i, TW_(i) waiting time of theexperiment i, G_(i) magnetic field gradient of the experiment i.

$\begin{matrix}{{f_{w,o,g}^{i}\left( {n,{T\; 2_{j}}} \right)} = {{HI}_{w,o,g} \cdot {\quad{{{\left\lbrack {1 - {\exp \left( {- \frac{{TW}_{i}}{{R_{w,o,g} \cdot T}\; 2_{j}}} \right)}} \right\rbrack \cdot {\exp \left( {- \frac{n \cdot {TE}_{i}}{T\; 2_{j}}} \right)} \cdot {\exp \left( {- \frac{n \cdot {TE}_{i}}{T\; 2\; D_{w,o,g}}} \right)}}\mspace{20mu} {T\; 2\; D_{w,o,g}}} = {{\frac{12}{D_{w,o,g} \cdot \gamma^{2} \cdot G_{i}^{2} \cdot {TE}^{2}}\mspace{20mu} i} \in \left( {1,\ldots \mspace{14mu},k} \right)}}}}} & (19)\end{matrix}$

The unknowns vector has three sets of bins, one for water, one for oiland other for gas. The range of three set of bins must cover all thepossible values of T2 and normally they will be overlapped, Equation 20.

$\begin{matrix}{B = {\begin{bmatrix}{BW}_{1} \\\vdots \\{BW}_{nBW} \\{BO}_{1} \\\vdots \\{BO}_{nBO} \\{BG}_{1} \\\vdots \\{BG}_{nBG}\end{bmatrix} = \begin{bmatrix}{BW} \\{BO} \\{BG}\end{bmatrix}}} & (20)\end{matrix}$

The apparent T2 spectrum is included in the data vector E and each T2has three corresponding intrinsic T2, related to different fluids (seeFIG. 7). Each bin of the apparent spectrum PP, in the tail of the datavector (Equation 21), is the sum of three different fluid bins at theintrinsic T2 value, according to Equation 22.

$\begin{matrix}{E = \begin{bmatrix}E_{1} \\\vdots \\E_{j} \\\vdots \\E_{k} \\{PP}\end{bmatrix}} & (21) \\{{T\; 2_{w,o,g}^{int}} = \frac{1}{\frac{1}{T\; 2} - \frac{1}{T\; 2\; D_{w,o,g}}}} & (22)\end{matrix}$

The A2I matrix with nB rows and nBW+nBO+nBG columns relates the apparentT2 spectrum to the vector of unknowns B, its rows have three ones, onefor each fluid, on the T2 given by Equation 22. If T2 does not have theexact value of some bin, it is divided between the two adjacent bins,according to the distance to them.

For each experiment i of k experiments and for each fluid, a matrixA_(w,o,g) ^(i) is defined by the equations 19 and 23.

$\begin{matrix}{A_{w,o,g}^{i} = {\quad\begin{bmatrix}{f_{w,o,g}^{i}\left( {1,{T\; 2_{1}}} \right)} & \ldots & {f_{w,o,g}^{i}\left( {1,{T\; 2_{j}}} \right)} & \ldots & {f_{w,o,g}^{i}\left( {1,{T\; 2_{{nB}_{w,o,g}}}} \right)} \\\vdots & \; & \ldots & \; & \vdots \\\ldots & \; & {f_{w,o,g}^{i}\left( {m,{T\; 2_{j}}} \right)} & \; & \ldots \\\vdots & \; & \ldots & \; & \vdots \\{f_{w,o,g}^{i}\left( {{nE}_{i},{T\; 2_{1}}} \right)} & \ldots & {f_{w,o,g}^{i}\left( {{nE}_{i},{T\; 2_{j}}} \right)} & \ldots & {f_{w,o,g}^{i}\left( {{nE}_{i},{T\; 2_{{nB}_{w,o,g}}}} \right)}\end{bmatrix}}} & (23)\end{matrix}$

The forward matrix A is formed by the lately defined sub-matrices,Equation 24.

Weighting is discussed next. An nB size diagonal matrix is added to theweighting matrix defined in the equation 14 giving Equation 25. K mustbe long enough to get a stable solution. 1_(B) is an identity matrix ofsize nB.

$\begin{matrix}{W = \begin{bmatrix}{\sigma_{1}^{- 2} \cdot 1_{1}} & 0 & 0 & 0 & 0 & 0 \\0 & \cdot & 0 & 0 & 0 & 0 \\0 & 0 & {\sigma_{j}^{- 2} \cdot 1_{j}} & 0 & 0 & 0 \\0 & 0 & 0 & \cdot & 0 & 0 \\0 & 0 & 0 & 0 & {\sigma_{k}^{- 2}{\cdot 1_{k}}} & 0 \\0 & 0 & 0 & 0 & 0 & {K \cdot 1_{B}}\end{bmatrix}} & (25)\end{matrix}$

No need for regularization and final result are discussed next. Noregularization is needed because the condition number of the matrix issmall enough with the proper K, Equation 26.

B=(A ^(T) ·W·A)⁻¹ ·W·A ^(T) ·E  (26)

B(1:nB_(w)) is the water intrinsic T2 spectrum,B(nB_(w)+1:nB_(w)+nB_(o)) is the oil intrinsic T2 spectrum andB(nB_(w)+nB_(o)+1:nB_(w)+nB_(o)+nB_(g)) is the intrinsic gas T2spectrum, the three corrected by hydrogen index. The total spectrum iscomputed summing bins with the same associated T2.

Adding conventional non-NMR curves to the inversion is discussed next.Assuming a total porosity petrophysical model only, resistivity logs donot respond to the matrix. They may be added without changes in theunknowns vector but to add other logs as density or acoustic the matrixvolume must be added to it, Equation 27.

$\begin{matrix}{{B = {\begin{bmatrix}{BW}_{1} \\\vdots \\{BW}_{nBW} \\{BO}_{1} \\\vdots \\{BO}_{nBO} \\{BG}_{1} \\\vdots \\{BG}_{nBG} \\V_{m}\end{bmatrix} = \begin{bmatrix}{BW} \\{BO} \\{BG} \\V_{m}\end{bmatrix}}}{{{\sum\limits_{i = 1}^{nBW}{BW}_{i}} + {\sum\limits_{i = 1}^{nBO}{BO}_{i}} + {\sum\limits_{i = 1}^{nBG}{BG}_{i}} + V_{m}} = 1}} & (27)\end{matrix}$

In the data vector, the square root of conductivity, other conventionalcurves and one representing the sum of all unknowns are added, Equation28.

$\begin{matrix}{E = \begin{bmatrix}E_{1} \\\vdots \\E_{j} \\\vdots \\E_{k} \\\sqrt{C} \\\rho \\{\Delta \; t} \\\vdots \\1 \\{PP}\end{bmatrix}} & (28)\end{matrix}$

The forward or evolution matrix thus becomes Equation 29.

Weighting for conventional logs is discussed next. The weighting factorsfor the conventional curves will be user defined parameters. The weightfor the sum of unknowns W_(S) is generally a large number, Equation 30.

$\begin{matrix}{\mspace{765mu} (30)} \\{W = {\quad{\quad\begin{bmatrix}{\sigma_{1}^{- 2} \cdot 1_{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & \cdot & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {\sigma_{j}^{- 2} \cdot 1_{j}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & \cdot & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & {\sigma_{k}^{- 2} \cdot 1_{k}} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & W_{C} & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & W_{D} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & W_{A} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & W_{S} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdot & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {K \cdot 1_{B}}\end{bmatrix}}}}\end{matrix}$

No regularization is applied but the factor K in the apparent T2spectrum plays a role in W other than the regularization factor.Equation 26 resolves the system. Penalty equations are additionalequations that minimize some value related to the unknowns. When penaltyequations are used to reduce the condition number of the system ofequations, the procedure is called regularization. A high conditionnumber characterizes ill conditioned systems. In this particular case,the penalty value is the norm of the vector of unknowns. To minimize thenorm as many penalty equations as unknowns are added to the system (β²·Iin equation 5 using matrix notation). The procedure is called minimumnorm regularization.

FIG. 10 is a flow chart for a method 100 for estimating one or moreproperties as a function of depth of an earth formation penetrated by aborehole. Non-limiting embodiments of the one or more propertiesincludes at least one selection from a group consisting of porosity,water volume, oil volume, gas volume, permeability, fluid viscosity,grain size and capillary pressure. Block 101 calls for receiving with aprocessor nuclear magnetic resonance (NMR) logging data having NMR echotrains as a function of depth in the borehole. The NMR logging data isobtained using an NMR tool conveyed through the borehole by a carrier.The NMR echo trains include NMR information such as a spectrum of T2values, diffusion constant D, and/or T1/T2 ratio R. Block 102 calls forreceiving with the processor non-NMR logging data having non-NMRmeasurement values for one or more types of non-NMR measurements as afunction of depth in the borehole. The non-NMR logging data is obtainedusing a non-NMR logging tool conveyed through the borehole by a carrier.Non-limiting embodiments of the non-NMR logging data include at leastone selection from a group comprising acoustic logging data, densitylogging data, neutron logging data, resistivity logging data and naturalgamma-ray logging data.

Block 103 calls for generating with the processor an evolution matrix(E) representing a mathematical relationship between the one or moreproperties in the property matrix (P) to be estimated and the NMRlogging data and non-NMR logging data matrix (M).

Block 104 calls for generating with the processor a matrix equation ofthe form M=E·P. In one or more embodiments, the matrix equation furtherincludes a term (ε) representing noise in a received signal with theform of the matrix equation being M=E·P+ε.

Block 105 calls for inverting with the processor the matrix equation toestimate the one or more properties as a function of depth. In one ormore embodiments, inverting includes obtaining a weighted least squaressolution of the matrix equation using a weighting matrix. In one or moreembodiments, the weighting matrix includes a diagonal matrix withelements equal to an inverse power of two of a standard deviation ofeach measurement.

The method 100 may also include inverting with the processor the matrixequation to estimate the one or more properties as a function of depth.In one or more embodiments, the hydrocarbon production-related actionincludes perforating a casing lining the borehole at a selected depthusing a perforation tool. In one or more embodiments, the hydrocarbonproduction-related action includes hydraulically fracturing the earthformation at a selected depth.

Embodiment 1. A method for estimating one or more properties as afunction of depth of an earth formation penetrated by a borehole, themethod comprising: receiving with a processor nuclear magnetic resonance(NMR) logging data comprising NMR echo trains as a function of depth inthe borehole; receiving with the processor non-NMR logging datacomprising non-NMR measurement values for one or more types of non-NMRmeasurements as a function of depth in the borehole; generating with theprocessor an evolution matrix (E) representing a mathematicalrelationship between the one or more properties in property matrix (P)to be estimated and the NMR logging data and non-NMR logging data matrix(M); generating with the processor a matrix equation of the form M=E·P;and inverting with the processor the matrix equation to estimate the oneor more properties as a function of depth.

Embodiment 2. The method according to any prior embodiment, furthercomprising performing a hydrocarbon production-related action usinghydrocarbon production-related apparatus and the one or more estimatedproperties as a function of depth.

Embodiment 3. The method according to any prior embodiment, wherein thehydrocarbon production-related action comprises perforating a casinglining the borehole at a selected depth using a perforation tool.

Embodiment 4. The method according to any prior embodiment, wherein thehydrocarbon production-related action comprises hydraulically fracturingthe earth formation at a selected depth.

Embodiment 5. The method according to any prior embodiment, wherein thematrix equation further comprises a term (ε) representing noise in areceived signal with the form of the matrix equation being M=E·P+ε.

Embodiment 6. The method according to any prior embodiment, wherein theone or more properties comprises at least one selection from a groupconsisting of porosity, water volume, oil volume, gas volume,permeability, fluid viscosity, grain size and capillary pressure.

Embodiment 7. The method according to any prior embodiment, wherein thenon-NMR logging data comprises at least one selection from a groupcomprising acoustic logging data, density logging data, neutron loggingdata, resistivity logging data and natural gamma-ray logging data.

Embodiment 8. The method according to any prior embodiment, whereininverting comprises obtaining a weighted least squares solution of thematrix equation using a weighting matrix.

Embodiment 9. The method according to any prior embodiment, wherein theweighting matrix comprises a diagonal matrix with elements equal to aninverse power of two of a standard deviation of each measurement.

Embodiment 10. A system for estimating one or more properties as afunction of depth of an earth formation penetrated by a borehole, thesystem comprising: a carrier configured to be conveyed through theborehole; a nuclear magnetic resonance (NMR) logging tool disposed onthe carrier and configured to provide NMR logging data comprising NMRecho trains as a function of depth in the borehole; a non-NMR tooldisposed on the carrier and configured to provide non-NMR logging datacomprising non-NMR measurement values for one or more types of non-NMRmeasurements as a function of depth in the borehole; and a processorconfigured to: receive the NMR logging data; receive the non-NMR loggingdata; generate an evolution matrix (E) representing a mathematicalrelationship between the one or more properties (P) to be estimated andthe NMR logging data and non-NMR logging data (M); generate a matrixequation of the form M=E·P; and invert the matrix equation to estimatethe one or more properties as a function of depth.

Embodiment 11. The system according to any prior embodiment, furthercomprising hydrocarbon production-related apparatus configured toperform a hydrocarbon production-related action using the one or moreestimated properties as a function of depth.

Embodiment 12. The system according to any prior embodiment, wherein thehydrocarbon production-related action comprises perforating a casinglining the borehole at a selected depth using a perforation tool.

Embodiment 13. The system according to any prior embodiment, wherein thehydrocarbon production-related action comprises hydraulically fracturingthe earth formation at a selected depth.

Embodiment 14. The system according to any prior embodiment, wherein thematrix equation further comprises a term (ε) representing noise in areceived signal with the form of the matrix equation being M=E·P+ε.

Embodiment 15. The system according to any prior embodiment, wherein theone or more properties comprises at least one selection from a groupconsisting of porosity, water volume, oil volume, gas volume,permeability, fluid viscosity, grain size and capillary pressure.

Embodiment 16. The system according to any prior embodiment, wherein thenon-NMR logging data comprises at least one selection from a groupcomprising acoustic logging data, density logging data, neutron loggingdata, resistivity logging data and natural gamma-ray logging data.

Embodiment 17. The system according to any prior embodiment, whereininvert comprises obtain a weighted least squares solution of the matrixequation using a weighting matrix.

Embodiment 18. The system according to any prior embodiment, wherein theweighting matrix comprises a diagonal matrix with elements equal to aninverse power of two of a standard deviation of each measurement.

In support of the teachings herein, various analysis components may beused, including a digital and/or an analog system. For example, the NMRtool 10, the non-NMR tool 15, the downhole production tool 21, thehydraulic fracturing system 24 and/or the controller 25 may includedigital and/or analog systems. The system may have components such as aprocessor, storage media, memory, input, output, communications link(wired, wireless, optical or other), user interfaces (e.g., a display orprinter), software programs, signal processors (digital or analog) andother such components (such as resistors, capacitors, inductors andothers) to provide for operation and analyses of the apparatus andmethods disclosed herein in any of several manners well-appreciated inthe art. It is considered that these teachings may be, but need not be,implemented in conjunction with a set of computer executableinstructions stored on a non-transitory computer readable medium,including memory (ROMs, RAMs), optical (CD-ROMs), or magnetic (disks,hard drives), or any other type that when executed causes a computer toimplement the method of the present invention. These instructions mayprovide for equipment operation, control, data collection and analysisand other functions deemed relevant by a system designer, owner, user orother such personnel, in addition to the functions described in thisdisclosure.

Further, various other components may be included and called upon forproviding for aspects of the teachings herein. For example, a powersupply (e.g., at least one of a generator, a remote supply and abattery), cooling component, heating component, magnet, electromagnet,sensor, electrode, transmitter, receiver, transceiver, antenna,controller, optical unit, electrical unit or electromechanical unit maybe included in support of the various aspects discussed herein or insupport of other functions beyond this disclosure.

The term “carrier” as used herein means any device, device component,combination of devices, media and/or member that may be used to convey,house, support or otherwise facilitate the use of another device, devicecomponent, combination of devices, media and/or member. Other exemplarynon-limiting carriers include drill strings of the coiled tube type, ofthe jointed pipe type and any combination or portion thereof. Othercarrier examples include casing pipes, wirelines, wireline sondes,slickline sondes, drop shots, bottom-hole-assemblies, drill stringinserts, modules, internal housings and substrate portions thereof.

The disclosure illustratively disclosed herein may be practiced in theabsence of any element which is not specifically disclosed herein.

Elements of the embodiments have been introduced with either thearticles “a” or “an.” The articles are intended to mean that there areone or more of the elements. The terms “including” and “having” and thelike are intended to be inclusive such that there may be additionalelements other than the elements listed. The conjunction “or” when usedwith a list of at least two terms is intended to mean any term orcombination of terms. The term “configured” relates one or morestructural limitations of a device that are required for the device toperform the function or operation for which the device is configured.

The flow diagram depicted herein is just an example. There may be manyvariations to this diagram or the steps (or operations) describedtherein without departing from the spirit of the invention. Forinstance, the steps may be performed in a differing order, or steps maybe added, deleted or modified. All of these variations are considered apart of the claimed invention.

While one or more embodiments have been shown and described,modifications and substitutions may be made thereto without departingfrom the spirit and scope of the invention. Accordingly, it is to beunderstood that the present invention has been described by way ofillustrations and not limitation.

It will be recognized that the various components or technologies mayprovide certain necessary or beneficial functionality or features.Accordingly, these functions and features as may be needed in support ofthe appended claims and variations thereof, are recognized as beinginherently included as a part of the teachings herein and a part of theinvention disclosed.

While the invention has been described with reference to exemplaryembodiments, it will be understood that various changes may be made andequivalents may be substituted for elements thereof without departingfrom the scope of the invention. In addition, many modifications will beappreciated to adapt a particular instrument, situation or material tothe teachings of the invention without departing from the essentialscope thereof. Therefore, it is intended that the invention not belimited to the particular embodiment disclosed as the best modecontemplated for carrying out this invention, but that the inventionwill include all embodiments falling within the scope of the appendedclaims.

What is claimed is:
 1. A method for estimating one or more properties asa function of depth of an earth formation penetrated by a borehole, themethod comprising: receiving with a processor nuclear magnetic resonance(NMR) logging data comprising NMR echo trains as a function of depth inthe borehole; receiving with the processor non-NMR logging datacomprising non-NMR measurement values for one or more types of non-NMRmeasurements as a function of depth in the borehole; generating with theprocessor an evolution matrix (E) representing a mathematicalrelationship between the one or more properties in property matrix (P)to be estimated and the NMR logging data and non-NMR logging data matrix(M); generating with the processor a matrix equation of the form M=E·P;and inverting with the processor the matrix equation to estimate the oneor more properties as a function of depth.
 2. The method according toclaim 1, further comprising performing a hydrocarbon production-relatedaction using hydrocarbon production-related apparatus and the one ormore estimated properties as a function of depth.
 3. The methodaccording to claim 2, wherein the hydrocarbon production-related actioncomprises perforating a casing lining the borehole at a selected depthusing a perforation tool.
 4. The method according to claim 2, whereinthe hydrocarbon production-related action comprises hydraulicallyfracturing the earth formation at a selected depth.
 5. The methodaccording to claim 1, wherein the matrix equation further comprises aterm (ε) representing noise in a received signal with the form of thematrix equation being M=E·P+ε.
 6. The method according to claim 1,wherein the one or more properties comprises at least one selection froma group consisting of porosity, water volume, oil volume, gas volume,permeability, fluid viscosity, grain size and capillary pressure.
 7. Themethod according to claim 1, wherein the non-NMR logging data comprisesat least one selection from a group comprising acoustic logging data,density logging data, neutron logging data, resistivity logging data andnatural gamma-ray logging data.
 8. The method according to claim 1,wherein inverting comprises obtaining a weighted least squares solutionof the matrix equation using a weighting matrix.
 9. The method accordingto claim 8, wherein the weighting matrix comprises a diagonal matrixwith elements equal to an inverse power of two of a standard deviationof each measurement.
 10. A system for estimating one or more propertiesas a function of depth of an earth formation penetrated by a borehole,the system comprising: a carrier configured to be conveyed through theborehole; a nuclear magnetic resonance (NMR) logging tool disposed onthe carrier and configured to provide NMR logging data comprising NMRecho trains as a function of depth in the borehole; a non-NMR tooldisposed on the carrier and configured to provide non-NMR logging datacomprising non-NMR measurement values for one or more types of non-NMRmeasurements as a function of depth in the borehole; and a processorconfigured to: receive the NMR logging data; receive the non-NMR loggingdata; generate an evolution matrix (E) representing a mathematicalrelationship between the one or more properties (P) to be estimated andthe NMR logging data and non-NMR logging data (M); generate a matrixequation of the form M=E·P; and invert the matrix equation to estimatethe one or more properties as a function of depth.
 11. The systemaccording to claim 10, further comprising hydrocarbon production-relatedapparatus configured to perform a hydrocarbon production-related actionusing the one or more estimated properties as a function of depth. 12.The system according to claim 11, wherein the hydrocarbonproduction-related action comprises perforating a casing lining theborehole at a selected depth using a perforation tool.
 13. The systemaccording to claim 11, wherein the hydrocarbon production-related actioncomprises hydraulically fracturing the earth formation at a selecteddepth.
 14. The system according to claim 10, wherein the matrix equationfurther comprises a term (ε) representing noise in a received signalwith the form of the matrix equation being M=E·P+ε.
 15. The systemaccording to claim 10, wherein the one or more properties comprises atleast one selection from a group consisting of porosity, water volume,oil volume, gas volume, permeability, fluid viscosity, grain size andcapillary pressure.
 16. The system according to claim 10, wherein thenon-NMR logging data comprises at least one selection from a groupcomprising acoustic logging data, density logging data, neutron loggingdata, resistivity logging data and natural gamma-ray logging data. 17.The system according to claim 10, wherein invert comprises obtain aweighted least squares solution of the matrix equation using a weightingmatrix.
 18. The system according to claim 17, wherein the weightingmatrix comprises a diagonal matrix with elements equal to an inversepower of two of a standard deviation of each measur